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A Generalized Linear Model (GLM) generalizes ordinary linear regression by allowing the response to follow a distribution in the exponential family, and modeling a transformed mean via a link function. The relationship is:
\[ g(\mathbb{E}[Y\mid X]) = X\beta, \]
where \(g\) is the link that connects the expected value of the outcome to the linear predictor \(X\beta\).
\[ g(\mathbb{E}[Y\mid X]) = X\beta. \]
Logistic regression is the most common GLM for binary outcomes. It links the log-odds of the outcome to predictors:
\[ \log\left(\frac{p}{1-p}\right) = X\beta, \]
where \(p = P(Y=1\mid X)\).
For individual binary outcomes (0/1), logistic regression assumes:
\[ Y \sim \text{Bernoulli}(p), \]
and if aggregated as counts out of \(n\), it is modeled as:
\[ Y \sim \text{Binomial}(n,p), \]
both under the same logit link.
A linear probability model can:
[0,1],Logistic regression bounds predicted probabilities and uses a proper likelihood framework.
A coefficient \(\beta_j\) represents the change in log-odds of the outcome per one-unit increase in \(X_j\), holding other variables constant.
The odds ratio for \(X_j\) is:
\[ \exp(\beta_j), \]
showing the multiplicative change in odds for a one-unit change in \(X_j\). Values \(>1\) raise odds; values \(<1\) lower odds.
A link function maps the expected outcome to the linear predictor. For the logit link with binary data:
\[ g(\mu)=\log\left(\frac{\mu}{1-\mu}\right), \]
which maps probabilities in \((0,1)\) to the real line.
GLMs are estimated using Maximum Likelihood Estimation (MLE). Parameters β are chosen to maximize the likelihood of the data, usually via iterative numerical methods such as Iteratively Reweighted Least Squares (IRLS).